Lecture 5 - Monday Week 3

The following notes were typed in \LaTeX during the lecture, and are full of typesetting errors. If someone has the time to check and correct them, please do.*

Vector Bundles

In the following, $M$ will be a smooth real manifold, but the same applies generally in other kinds of manifolds. Also, we will use $k$ to represent the field $\R$ or $\C$.

Definition of Vector Bundle

Let $M$ be a smooth manifold. A rank $r$ vector bundle $E$ over $M$ is

A map $\pi: E -> M$ smooth map between manifolds,

$\forall m \in M$, the fibre $\pi^{-1}(m)$ is a $k-$vector space of rank $r$,

$E$ is locally trivial: i.e. $\exists$ open cover $U_\alpha$ of $M$, and map $\chi_\alpha:\pi^{-1}(U_\alpha) \rightarrow U_\alpha \times k^r$ which respects the vector space structure, i.e. it is fiber-wise a vector space isomorphism.

def: $E$ is trivial if $E \cong M \times k^r$.

examples: $TM, T^*M, \wedge^k T^*M$, tensor bundles. You get all these if you have a smooth manifold. These do not arise naturally from topological manifolds though for example. Local trivialisations of these come from the coordinate patches for $M$.

example: An example of a non-trivial vector bundle is the Moebius strip where you extend all fibres to the real line. this is an example of a \textbf{line bundle}, i.e. a rank $1$ vector bundle.

From local trivialisations: $\pi^{-1}(U_\alpha \cap U_\beta) \to^{id} \pi^{-1}(U_\alpha \cap U_\beta) = \chi_\beta \circ \chi_\alpha^{-1}$... INPUT DIAGRAM HERE

We can encode this into "transation funtions", smooth maps $\rho_{\beta\alpha}: U_\alpha \cap U_\beta \to GL(k, r)$ where $\chi_\beta \circ \chi_\beta^{-1} (m, v)\mapsto (m, (\rho_{\beta\alpha})(v))$.

Transition functions satisfy fact that :

$\rho_{\beta\alpha} = \rho_{\alpha\beta}^{-1}$ (*)

$\rho_{\gamma\alpha} = \rho_{\gamma\beta} \circ \rho_{\beta\alpha} $ on $U_\alpha \cap U_\beta \cap U_\alpha$ (**)

or, $\rho_{\alpha\gamma} \circ \rho_{\gamma\beta} \circ \rho_{\beta\alpha}$ (cocycle condition).

Vice versa, given an open cover $U_\alpha$ of $M$ and transtion functions $\rho_{\beta\alpha}$ satisfying (*) and (**) we can build a vector bundle $E = \coprod U_\alpha \times k^r / ~$ where $(m, v) \in (U_\alpha \cap U_\beta \times k^r) \subset (u_\alpha \times k^r)$ is equivalent to $(m, w) \in (U_\alpha \cap U_\beta \times k^r) \subset (u_\beta \times k^r)$ if $w = \rho_{\beta\alpha}(v)$.

Claim: $E$ is a smooth vector bundle over $M$. (reconstruction theorem).

def: Morphisms of vector bundles: is a map $\Phi: E -> F$ such that there exists a map $\phi: M -> N$ such that the diagram ... commutes, and $\Phi$ is linear on all fibres of $\pi_E$ and $ \pi_F$.

def: If $M = N$ and $\phi = id_M$, then $\Phi$ is "vertical".

example: If $f: M \to N$ is a smooth map between manifolds then $df = f_* : TM -> TN$ is a vector bundle morphism.

\subsubsection {Pullbacks}

If you have a vector bundle over a manifold, and a smooth map from another manifold to it's base, you get a map ..

If $E \to^\pi N \leftarrow^f M$ is a vector bundle $E$ and smooth map $f$, then we get maps $M \leftarrow f^*E \rightarrow E$ such that the right arrow is a bundle morphism that is a fibre-wise isomorphism. It always exists. We can construct it via the reconstruction theorem: Pick local trivialisation $U_\alpha, \chi_\alpha$ of $N$ with transition functions $\rho_{\beta\alpha}$. Then $f^{-1}(U_\alpha)$, use $\rho_{\beta\alpha}$ as transition functions.

Sections and Frames

Given a vector bundle $E \to^\pi M$, a section $s$ of $E$ is a smooth map $M \to^s E$ that is a one-sided inverse of $\pi$. i.e. $\pi \circ s = id_M$.

example: vector fields are sections of the tangent bundle. example: differential k-forms are sections of the $k$th exterior bundle of the cotangent bundle $\wedge^k T^*M$. example: tensor fields are sections of some tensor bundle.

def: $\Gamma(E)$, the space of sections of $E$ is an infinite-dimensional $k-$vector space. Note this only holds in the $C-\infty$ setting, and doesnt hold in the topological or analytic setting.

def: A frame for $E$ is a collection of $r$ sections that are linearly independent fibre-wise.

prop: frames $\leftrightarrow$ trivialisations of $E$.

cor: frames always exist locally, but not globally.